A bug and a grasshopper enjoy a ride on a spinning rod that rotates freely about a vertical axle withangular velocity @o. The bug has mass m¡, the grasshopper has mass m,, and the rod has moment ofinertia IR about the axis of rotation. Each of the insects is at a distance S from the axle. Suddenly thebug and the grasshopper see an approaching bird. In desperation, the grasshopper jumps offhorizontally in the direction shown (see the top view figure) with velocity of magnitude vo, while thebug falls off the rod vertically down. What is the angular velocity (magnitude and direction) of the rodafter both insects have left the rod?Top viewахisVo

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Asked Dec 9, 2019
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Need help on this problem. See attatched picture.

A bug and a grasshopper enjoy a ride on a spinning rod that rotates freely about a vertical axle with
angular velocity @o. The bug has mass m¡, the grasshopper has mass m,, and the rod has moment of
inertia IR about the axis of rotation. Each of the insects is at a distance S from the axle. Suddenly the
bug and the grasshopper see an approaching bird. In desperation, the grasshopper jumps off
horizontally in the direction shown (see the top view figure) with velocity of magnitude vo, while the
bug falls off the rod vertically down. What is the angular velocity (magnitude and direction) of the rod
after both insects have left the rod?
Top view
ахis
Vo
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A bug and a grasshopper enjoy a ride on a spinning rod that rotates freely about a vertical axle with angular velocity @o. The bug has mass m¡, the grasshopper has mass m,, and the rod has moment of inertia IR about the axis of rotation. Each of the insects is at a distance S from the axle. Suddenly the bug and the grasshopper see an approaching bird. In desperation, the grasshopper jumps off horizontally in the direction shown (see the top view figure) with velocity of magnitude vo, while the bug falls off the rod vertically down. What is the angular velocity (magnitude and direction) of the rod after both insects have left the rod? Top view ахis Vo

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Expert Answer

Step 1

Let mr denote the mass of the rod and let the length of the rod be denoted as l.

 The moment of inertia IR of the rod when the axis of rotation is at through the its center perpendicular to its length is

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,]? т, 12 m,S -;l = 2S 3

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Step 2

The law of conservation of angular momentum of a system states that,

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L, = L;

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Step 3

Here Li and Lf are the initial and final angular momentum of the system respectively.

Let the moment of inertia of the bug and the grasshopper about the axis of rotation be denoted by IB ...

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L, = (I +I, +IG)@, m,s? - + m,s² +m,s² |@, 3

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